Introduction
The material below explains the concepts of exponential growth and doubling time - the time it would take a population to double if it were growing exponentially at a constant rate. Throughout human history, human population has generally grown in an exponential manner, and discussion of exponential doubling times has been appropriate (see chart at the end of this page).Now, the dynamics of population growth are more complex and growth can no longer be considered truly exponential. The issues of the 21st Century are migration, aging, the youth bulge, urbanization, and all of the very new and very serious socio-economic-political consequences, as well as continued growth in the developing world and, of course, pressures on resources and the environment - global and local.
This does not mean that countries will not double their population. Many will. For example, the U.S. is projected to double its population this century, practically within the lifetimes of today's children. So it is still appropriate to reference the time in which these countries will double their population, which is perhaps the most illuminating manner of presenting population growth to the lay person. But it should be noted that the explanations below refer to a constant rate of growth, which is not as applicable to countries now as in the past.
Exponential Growth
A quantity grows exponentially when its increase is proportional to what is already there. A common example is compound interest, where $100 invested at 7% per year annual compound interest will double in 10 years! Exponential growth applies to populations, too -- if a population grows at 7% per year, it, too, will double in 10 years.There are surprising consequences to the phenomenon of exponential growth. The $100 invested at a 7% annual return will double in 10 years to approximately $200, double in another 10 years to approximately $400, and double again in the next 10 years to approximately $800. Significant gains can be made by simply relying on exponential growth over time. One way of saying this is that the longer you wait (on your investment), the faster your returns come in. In the following graph, you can see that over time, returns increase dramatically.
Unfortunately, exponential growth works against us, too. When populations continue to grow, the impact of growth becomes increasingly significant over time. In other words, because of the nature of exponential growth, "when things get bad, they get bad in a hurry".
Consider a country with 100 people, growing at 7% per year. In 10 years, the population will double to 200 people, in another 10 years it will double again to 400 people, and ten years after that it will double again to 800 people. The following graph shows this exponential population growth. Notice the shape of the curve is the same as in the graph above!
The Lily Pond Parable
- If a pond lily doubles everyday and it takes 30 days to completely cover a pond, on what day will the pond be 1/4 covered?
- 1/2 covered?
- Does the size of the pond make a difference?
- What kind of environmental consequences can be expected
as the 30th day approaches?
- What will begin to happen at one minute past the 30th day?
- At what point (what day) would preventative action become necessary to prevent unpleasant events?
- With respect to human population, what corresponding day are we at in the world? The United States?
The Rule of 70
The Rule of 70 is useful for financial as well as demographic analysis. It states that to find the doubling time of a quantity growing at a given annual percentage rate, divide the percentage number into 70 to obtain the approximate number of years required to double. For example, at a 10% annual growth rate, doubling time is 70 / 10 = 7 years.Similarly, to get the annual growth rate, divide 70 by the doubling time. For example, 70 / 14 years doubling time = 5, or a 5% annual growth rate.
The following table shows some common doubling times:
| Growth Rate (% per Year) |
Doubling Time in Years |
|---|---|
| 0.1 | 700 |
| 0.5 | 140 |
| 1 | 70 |
| 2 | 35 |
| 3 | 23 |
| 4 | 18 |
| 5 | 14 |
| 6 | 12 |
| 7 | 10 |
| 10 | 7 |
Math Behind the Rule of 70
The use of natural logs arises from integrating the basic differential equation for exponential growth: dN/dt = rN, over the period from t=0 to t = the time period in question, where N is the quantity growing and r is the growth rate.- The integral of that equation is:
- N(t) = N(0) x ert
where N(t) is the size of a quantity after t intervals have elapsed, N(0) is the initial value of the quantity, e is the base of the natural logarithm, r is the average growth rate over the interval in question, and t is the number of intervals. Natural logarithms (that is, logarithms to the base e) come in from this integration. Natural logs are sometimes abbreviated ln to distinguish them from "common" logarithms of base 10.
In our case, t is usually given in years and r is the average annual growth rate. However, the formula works for seconds, weeks, centuries, etc.
The ratio of the final to the initial values of N is N(t)/N(0), and is equal to ert power. (Dividing through by N(0).)
The natural logarithm of the ratio N(t)/N(0) is equal to rt. (Taking the natural log of e raised to a power is simply the power itself.)
The growth rate r is then given by natural log N(t)/N(0) divided by t.
- Alternatively, if one knows the final and initial values of N and the
average growth rate, one can find the time it takes at that average
growth rate for the quantity to grow from its initial value to the final
value.
- t = ln [ N(t)/N(0) ] / r
- A special case is the doubling time, which is the time when N(t)/N(0) =
2, that is the quantity has doubled from its initial value. At that
point
- rt = ln 2 = 0.69
If one knows the growth rate as a decimal fraction, then the doubling time t2 = 0.69 / r.
If the growth rate is given in percent, then 0.69 must be multiplied by 100, and the doubling time = 69/r. This is the origin of the rule of 70, i.e., 69 is rounded up to 70.
For the results to be accurate, all of these calculations assume that the growth rate remains unchanged throughout the interval in question, that is, that the growth is exponential at the average rate for the entire period. Most quantities don't really grow that way (compound interest is an exception), so this method is generally an approximation for the real world.
When considering growth over a period of years, it is important to note that taking the natural logarithm of the ratio of the final value to the initial value and dividing by the time period in years gives the average annual growth rate.
For example, metro Denver population grew 15% from 1990 to 1996. The simple arithmetic average of growth is 2.5% per year (15% / 6 years = 2.5%/year). Strictly speaking the rule of 70 applies to exponential growth, which means that the compound average population growth rate must be divided into 70 to get the doubling time.
The compound average growth rate involves natural logarithms. To find the compound growth rate, take the natural log of the ratio of the final to the initial populations and divide this by 6 years. 15% growth means that the ratio of final to initial populations is 1.15. (The final population is 115% of the initial population, considered to be 100%; 115% / 100% = 1.15.) The natural log of 1.15 is 0.14. Dividing 0.14 by 6 years = 0.023/year, or an average geometric increase of 2.3% / year.
Finally, divide 70 by 2.3 to give a doubling time of 30 years.
-- Mathematical explanation by Dick Schneider
Additional References
Algebra the Easy Way, Douglass Downing; 1996, ISBN 0812093933, (334p, $11). Includes a basic discussion of exponential functions and logarithms.
Ask. Dr. Math
Beyond the Limits, Confronting Global Collapse, Envisioning a Sustainable Future, Donella Meadows. The first two chapters present an excellent discussion of exponential growth.
Doubling Times and the Rule of 70, Mathematical Methods for Economics, Michael Klein.
Is There A Population Problem?, Albert A. Bartlett.
Population Doubling Time, Population-Environment Balance.
Answers to The Lily Pond Parable
- If a pond lily doubles everyday and it takes 30 days to completely cover a pond, on what day will the pond be 1/4 covered?
- Answer: Day 28.
Growth will be barely visible until the final few days.
(On the 25th day, the lilys cover 1/32nd of the pond;
on the 21st day, the lilys cover 1/512th of the pond).
- 1/2 covered?
- Answer: Day 29.
- Does the size of the pond make a difference?
- Answer: No. The doubling time is still the same. Even if you could magically double the size of the pond on day 30, it would still hold only one day's worth of growth!
- What kind of environmental consequences can be expected
as the 30th day approaches?
- Answer: The pond will become visibly more crowded each day, and this crowding will begin to exhaust the resources of the pond.
- What will begin to happen at one minute past the 30th day?
- Answer: The pond will be completely covered. Even though the lilys will be reproducing, there will be no more room for additional lilys, and the excess population will die off. In fact, since the resources of the pond have been exhausted, a significant proportion of the original population may die off as well.
- At what point (what day) would preventative action become necessary to prevent unpleasant events?
- Answer: It depends on how long it takes to implement the action and how full you want the lily pond to be. If it takes two days to complete a project to reduce lily reproductive rates, that action must be started on day 28, when the pond is only 25% full -- and that will still produce a completely full pond. Of course, if the action is started earlier, the results will be much more dramatic.
- With respect to human population, what corresponding day are we at in the world? The United States?
"Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist" --Kenneth Boulding
Copyright 1995-2024 Fred Elbel. This material may be freely used and distributed only for non-commercial purposes, with credit. Trademarks and copyrighted items remain the property of the owner.